Can the determinant of a combinatorial matrix be proven to always equal 1?

[rbzima]

Let A\Large be the n \times n matrix (a_{ij}) given by
a_{ij} = \binom{m_j + i - 1}{j - 1}
where i,j = 1, 2, ..., n and n and m_j are natural numbers. Find the determinant of A\Large.

So, I've been looking at this problem for the past couple days and it is a really interesting problem. I discovered that the determinant is actually 1, however I'm wondering how to show this result in terms of a proof. If anyone has any suggestions, they would be greatly welcomed! ;)

I personally was thinking that when you transpose the matrices, something interesting happens with the nested matrices in the upper left corner moving outward, but I'm not entirely sure this is a right approach to be taking. Also, finding determinants by cofactors is really only helpful in smaller matrices, so I was wondering if it might also need to be expanded using the Big Formula, which in this case would consist of n! different terms.

 

[fresh_42]

I would first check whether it is still true for all $m_j =1$. Then an induction over $n$ should be the way to prove it. Also formulas for the various diagonals might help a lot.